Range Estimation Under Constraints Is Computable Unless There Is a Discontinuity

Abstract

One of the main problems of interval computations is computing the range of a given function over given intervals. It is known that there is a general algorithm for computing the range of computable functions over computable intervals. However, if we take into account that often in practice, not all possible combinations of the inputs are possible (i.e., that there are constraints), then it becomes impossible to have an algorithm which would always compute this range. In this paper, we explain that the main reason why range estimation under constraints is not always computable is that constraints may introduce discontinuity—and all computable functions are continuous. Specifically, we show that if we restrict ourselves to computably continuous constraints, the problem of range estimation under constraints remains computable.